THE SEVENTEEN RESISTANCES OF THREE RESISTORSBy N. W. LEWIS, Ph.D.(Eng.), Member.*

{The paper was first received J8/A December, 1944, and in revised form \5th March, 1945.)

SUMMARYThree resistors offer 4 series, 4 parallel, and 6 series-parallel com-

binations, a total of 17 resistances. It is shown that if the values ofthe resistors are three consecutive terms of a certain geometric series(common ratio approximately 11/6) then the distribution of the 17values is closely related to another geometric series (common ratioapproximately 6/5). The results apply equally to the combinationsof three inductors or three capacitors.

Seventeen different values of resistance may be obtained fromthree resistors, including all the series, parallel, and series-parallel combinations. This simple fact must often have givenrise to the question whether a relation between the three resistorscan be found to yield a distribution of the 17 values in accordancewith some simple law, e.g. an arithmetic or geometric series.The question applies equally to the combinations of three in-ductors or three capacitors. It is shown in the following notesthat a distribution close to a geometric series may be obtained.This, believed to be the simplest possible solution, seems ofsufficient interest and practical value to justify its being set forth,the author being unable to find any published reference to thesubject.

Briefly, the result is that if the values of the resistors are threeconsecutive terms of the geometric series whose common ratio isa certain quantity which will be termed A and is approximately

Table 1

ValueNo

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1

1/(A -

1/(A-

A(A -

A

M(A -

A2(A -

A2(A -

A2

A2/(A -

A2/(A -

A 3 ( A -

A3

A3/(A -

A 4 ( A -

A 4 ( A -

A4

1) ==

1)2 =,

1) =

1) -

1)2 =

1) ,-,

-.:

- 1) ^

I ) 2 -

1) -

- 1) =

1)2 ==

1) -

=

Value

1 =

a =

1 + l/2a =--

2a - I/a =

1 + I/a =

a + 12a

2 + I/a =

2a + 1 =

a + 2 + I/a -

2a + 2 + l/2a =

2a + 2 + l/« =

2a + 3 + I/a =

3a + 3 + I/a =

2a + 4 + 2/a =

4a + 4 + I/a =

4a + 5 + 2/a =

10000

11915

1-4196

1•5437

1-8393

21915

2-3830

2-8393

3-3830

4-0308

4-8026

5-2223

6-2223

7-4138

80616

9-6053

11-4446

Ratio ofconsecutive

values

a = 1-1915

a = 11915

0= 1-0874

a = 11915

a = 11915

/? = 1-0874

a = 1-1915

a== 1-1915

a = 11915

a = 1-1915

p= 1-0874

a - 1-1915

a = 11915

/ ? = 10874

a - 1-1915

<x= 1-1915

11/6, then the 17 values will be closely related to another geo-metric series whose common ratio is a quantity a, approximately6/5. Of the 16 ratios between consecutive values, it is foundthat 12 are precisely the quantity a, and the remaining four areeach a quantity B which, by what appears to be a curious quirkof nature, is within 0-4% of the square root of a.

Another curious fact is that a is within 0-2% of the fourthroot of 2, so that plotting the 17 values on a logarithmic scalegives, approximately, 12 quarter-octave and 4 eighth-octaveintervals, covering 3£ octaves in all.

Table 1 shows the 17 values (based on a minimum of unity)and their ratios, and Fig. 1 shows the connections of the com-binations in the same ascending order of value. It will be seenthat values Nos. 5, 9 and 13 are those of the three resistors taken

VALUE

2

3

4

5

6

7

8

9

10

n

12

13

14

15

16

17

RESISTORS(OR INDUCTORS)

m

at *3

A>2 * 3

R2

CAPACITORS

ci a a

cr c

cr ci

C2

C1

HI-C2 Ci

CI C2

• Post Office Engineering Department.

[378 ]Fig. i

LEWIS: THE SEVENTEEN RESISTANCES OF THREE RESISTORS 379

singly, and that values Nos. 1 and 17 are two more terms in thesame A-series.

The A-series suggested itself as a development of the "^-series,"which is perhaps best known to students of artistic compositionand is related to the Fibonacci Numbers of 13th century origin.The ^-series is the geometric series whose common ratio <j> issuch that each term is the sum of the two preceding terms. (f> isthus a root of the equation

whence, taking the positive root cf> = i(l + V5) = 1-618If the values of two resistors are consecutive terms of a ^-series,then the values of the parallel and series connections of the re-sistors are respectively the preceding and succeeding terms ofthe same series.

For the purpose of these notes, the Fibonacci Numbers maybe considered as the series of integers (commencing with 1, 2) inwhich each term is the sum of the two preceding terms, namely

1 2 3 5 8 13 21 34 55 . . .

The ratios of consecutive Fibonacci Numbers are approxima-tions to <j), becoming closer with ascending order of number.

The A-series is analogous to the (^-series and maybe denned asthe geometric series whose common ratio A is such that eachterm is the sum of the three preceding terms. A is thus a rootof the equation

A3 - A2 - A - 1 = 0

whence A = i [ l + (19 + 3V33)1/3 + (19 - 3V33)1/3] = 1 • 83929One of the odd properties of the A-series is that A, A2 and A3

are respectively the harmonic, geometric and arithmetic meansof 1 and A4.

It may be seen from the relationships given in Table 1 that

a = 1/(A- 1 ) = 1-19148

Table 2

and = 2/A= 1-08738

It may also be noted that a 3 . jS = A, and a3 . j92 = 2.The related number series, analogous to the Fibonacci

Numbers, may be defined as the series of integers (commencingwith 1, 2, 3) in which each term is the sum of the three precedingterms, viz.

1 2 3 6. 11 20 37 68 125 . . .

The ratios of consecutive terms are approximations to A, be-coming closer with ascending order of number. The ratios 11 /6and 20/11 are sufficiently close to A for the 6-11-20 combinationand its multiples to be convenient for. many practical purposes.For example, resistors of 18, 33 and 60 ohms conveniently covera decade range in steps not exceeding a 6/5 ratio, as shown inTable 2.

As a matter of practical interest, a method of selecting any ofthe 17 combinations by means of U-links and sockets is shownin Fig. 2. This is one of several possible arrangements requiring

Value No.

1

2

3

4

5

6

7

8

9

10

11

12

*13'

1 4 > * • • •

15

16

17

Value

9-754

11-647

13-846

15081

18000

21-290

23-189

27-568

33000•~"~^™~~"

39-290

46-846

51000

60-000

71-647

78-000

93000

111000

Ratio ofconsecutive

values

1-1941

1-1888

10892

1-1935

1-1828

1-0892

1-1888

11971

11906

1•1923

10887

11765

11941

10887

11923

1-1935

Fig. 2

only 8 sockets. Each combination is obtained by the appropriateinsertion of one, two, three or four U-links, each link making aconnection between a pair of adjacent sockets. Another method,requiring a 17-way 4-bank rotary switch, is shown in Fig. 3.

Ri

R2

R3f-WWV-4

6 0 0 0 0 0 0 0 0 0 6 0 0 o o o o

0 6 0 0 0 0 6

1 j j 1 i . 0 0 0 0 0 6 0 0 0

j 1 1 1 . J 1 J J 1 . . 1TTT1t 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17

Fig. 3